Questions tagged [principal-bundles]
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product $X\times G$ of a space $X$ with a group $G$.
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Is the change of trivialisations on a principal $G$-bundle given above each basepoint by a right-multiplication with a group element?
Let $\pi: M \rightarrow B$ be a principal $G$-bundle. Suppose $(U_i, \phi_i)$, $(U_j, \phi_j)$ are two (equivariant) trivialisations $\phi_i: \pi^{-1}(U_i) \rightarrow U_i \times G$ (similarly for $\...
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Isomorphism of real vector spaces $\Omega^1(\mathfrak u(n)) \cong \Omega^{0,1}(\mathfrak{gl}(n,\mathbb C))$
Consider a compact Riemann surface $X$. In [1, p. 570] the isomorphism of real vector spaces
$$
\Omega^1_{\mathbb R}(X,\mathfrak u(n)) \cong \Omega^{0,1}(X,\mathfrak{gl}(n,\mathbb C))
$$
is mentioned. ...
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Induced action on vector valued differential forms
Let $F(X)$ be the frame bundle of a smooth $n$-dimensional manifold $X$. Let $G$ be a finite subgroup of $GL_n(\mathbb C)$ acting by automorphisms on $X$ and let $V$ be a $GL_n(\mathbb C)$-...
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Locally Compact Lie Groups and Matrix Lie Groups
Let $G$ be any Lie group with a Lie algebra $\mathfrak{g}(n, \mathbb{R})$; but assume that I only deal with a locally compact, connected (topologically) component around its $e \in G$ element, say $K \...
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Yang-Mills action is invariant under conformal change of the metric
Let $(M,g)$ be a pseudo-Riemannian 4-manifold with a principal bundle $P\to M$. Prove that the Yang–Mills action $S_{YM}[A]$ is invariant under a conformal change of the metric $g$: $\quad g'=\mathrm{...
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Covariant derivative defined by Maurer-Cartan form
Let $G$ be a matrix Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a closed Lie subgroup with Lie algebra $\mathfrak{h}$ embedded in $\mathfrak{g}$.
Suppose we have chosen a complementary ...
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Can I assume a horizontal vector is a horizontal lift?
I am reading a proof that the curvature 2-form is given by $(F_A)_p(v,w) = dA_p(v,w) + [A_p(v),A_p(w)]$ where $A\in \Omega^1(P;\mathfrak{g})$ is a connection 1-form on $P\xrightarrow{\pi}M$.
One of ...
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Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$?
Let $S$ be a $G$-structure on $M$ and let $\nabla$ be a connection on $M$ compatible with $S$, i.e. the parallel transport preserves $S$.
Consider now the induced principal bundle connection on $Fr(TM)...
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Vector bundle associated to the universal cover $\mathbb{R}\to S^1$
It's a well known fact that, given a principal $G$-bundle (where $G$ is a Lie subgroup of $\text{GL}(r,\mathbb{R})$)
$$\pi_P:P\to X$$
there is an associated vector bundle
$$\pi_E:E(P):=(P\times \...
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Why is the curvature a horizontal form?
Let $\pi:P\to M$ be a principal $G$-bundle, let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\omega\in\Omega^1(P,\mathfrak{g})$ be a connection form, i.e.
$$R_g^*\omega=\text{Ad}_{g^{-1}}\omega\ \...
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Is there a simple description of the total space of a principal S^1 bundle over a compact surface?
It is known that principal $S^1$-bundles over a compact surface $\Sigma_g$ are classified by their Chern classes in $H^2(\Sigma_g, \mathbb{Z}) \cong \mathbb{Z}$.
When the Chern number is zero, the ...
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Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?
Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
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Connection on $U(1)$-bundle vs. 1-form
My original idea of a $\mathfrak{u}(1)$-valued connection $\omega$ is that it's simply a normal $1$-form. But in the middle of page 3 of https://www.arxiv.org/abs/math/0511710 he says "a ...
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What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?
Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
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Derivation of a spin connection in general relativity
On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle.
The ...
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